Add butter, powdered sugar, salt, and vanilla extract to the bowl of a large food processor. Process until smooth, about eight 5-second pulses. Add flour and pulse until the mixture looks crumbly and coarse, like sand, about six 5-second pulses. Add sprinkles and quickly pulse several more times, until the sprinkles are just incorporated. Overworking them will make them bleed.
Turn the dough out onto a work surface and shape it into two logs, about 2 inches in diameter. Tightly wrap each log in plastic wrap and freeze for 30 minutes.
Preheat oven to 350°F. Line two baking sheets with parchment paper or a silicone baking mat. Remove one log of dough from the freezer. Cut dough into ¼-inch to ⅓-inch thick slices, rolling log every few slices to maintain the round shape of the cookies. Arrange cookies on the prepared baking sheet. Bake until cookies just barely turn a light golden color around the edges, between 10 and 15 minutes.
Allow to cool on baking sheets for 5 minutes before removing to a cooling rack to cool completely before storing in an airtight container.
Repeat with the second cookie dough log.
Notes
Unbaked cookie dough can be refrigerated for up to 5 days.